The England under-19's striker makes switch to League Two side after the signing of Daniel Sturridge from Chelsea pushes him further down the pecking order at Anfield
Brendan Rodgers' is sending the 18-year-old to the League Two side to pick up experience after featuring briefly for the Reds' this season.
Morgan started three of the Merseyside outfit's Europa League games this campaign, against Hearts twice and Anzhi Makhachkala.
"A lot of hard work has taken place to get this young lad to the club," United manager Steve Evans told the club's official website
The striker, who has played for England at Under-17 and Under-19 level, will stay at the Millers until February 9.
Morgan will join up with Evans' squad ahead of their trip to face Aldershot in the third round of the FA Cup.